Perturbative calculation of the spin-wave dispersion in a disordered double-exchange model

a r X i v :c o n d -m a t /0406134v 1 [c o n d -m a t .s t r -e l ] 5 J u n 2004

Typeset with jpsj2.cls Letter

Perturbative calculation of the spin-wave dispersion in a disordered

double-exchange model

Taeko Semba and Takahiro Fukui

Department of Mathematical Sciences,Ibaraki University,Mito 310-8512,Japan

(Received February 2,2008)

We study the spin-wave dispersion of localized spins in a disordered double-exchange model using the perturbation theory with respect to the strength of the disorder potential.We calculate the dispersion upto the next-leading order,and extensively examine the case of one-dimension.We show that in that case,disorder yields anomalous gapped-like behavior at the Fermi wavenumber of the conduction electrons.

KEYWORDS:colossal magnetoresistance manganites,double-exchange model,randomness,

spin-wave dispersion,perturbation

Colossal magnetoresistance manganites 1–3have been providing hot topics owing to their rich magnetic and electronic properties.4–6Recent studies have clarified that the double-exchange (DE)mechanism 7–10alone is not enough to understand a variety of phases of these materials,which is actually induced by the interplay between spin,charge and orbital degrees of freedom.

Recently,the softening of the spin-wave dispersion,11–15observed in R 1−x D x MnO 3(R=La,Pr,Nd and D=Ba,Sr,Ca,etc.),has suggested that a random potential also plays a crucial role in magnetic properties of these materials.16–21Although some other mechanism 22–26have been proposed to describe the softening phenomena,recent numerical calculations by Motome and Furukawa 27–29have shown that a simple DE model with a random potential can explain the anomalous behavior of the spin-wave spectrum including broadening,gap formation as well as softening.

Motivated by their work,we study a disordered DE model by the use of the perturbation with respect to the strength of a random potential.Based on a general formula we derived,we show that in the case of 1-dimensional (1D)systems randomness yields a singularity of the spin-wave dispersion at the Fermi wavenumber of conduction electrons.It turns out that such anomalous behavior is mainly due to the nesting of the Fermi surface in 1D models.We then discuss the higher dimensional systems,based on the results of 1D systems.

We start with the following Hamiltonian:

H =−t i,j σ

c †

iσc jσ−J H

i

s i ·S i + i

ǫi c †iσc iσ,(1)

where s i =

1

the spin-S representation,and ǫi is the quenched random potential with statistical properties

ǫi ǫj =gδi,j .This model includes two large parameters,S and J H .Assuming that

localized spins are almost aligned to the z -direction,we first take into account the leading terms with respect to the series of 1/S .To this end,we utilize the Holstein-Primakoffmapping for localized spins:

S +i

=

√

2Sa †i ,S z

i =S −a †i a i ,

(2)

which leads us to the following truncated Hamiltonian up to O (1/S 0):

H =−t i,j ,σ

c †iσc jσ−

J H S

S

2

i,σ

a †i a i σc †

iσc iσ+

i,σ

ǫi c †iσc iσ.

(3)

Based on this Hamiltonian,we calculate the spin-wave dispersion perturbatively with respect to the strength of disorder g ,collecting the leading terms with respect to J H .

Previously,Furukawa 30calculated the magnon dispersion in the clean DE model,according to the diagrams (a)in Fig.1.In the presence of disorder,these diagrams should be evaluated by the use of the electron propagator with the self-energy induced by the scattering of electrons by impurities.Since down-spin electrons are located in the momentum space far above the Fermi energy in the large J H limit,it is readily seen that disorder yields the self-energy to up-spin electrons only,which is evaluated as

Σ↑(iεn )=

k

−g

τ

sgn(εn )(4)

in a weak disorder limit,where εn =(2n +1)π/βis the fermionic Matsubara frequency,

εk =−2t

µcos k µ−εF is the bare electron dispersion,and the relaxation time τis given by 1/τ=πgD (εF

).

=+

+ ...

=+

q

(a)

(b)Fig.1.(a)The leading self-energy of the spin-wave.Thin-lines denotes the down-spin propagator,

whereas the thick-line denotes the up-spin propagator with the self-energy induced by disorder.(b)The propagator of up-spin electrons used in the diagrams (a).The thin-line and dotted-line denotes the bare up-spin propagator and impurity potential,respectively.